Solve the Absolute Value Inequality for y |y|>=4.5

$| y | \geq 4.5$

Write $| y | \geq 4.5$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$y \geq 0$

In the piece where $y$ is non-negative, remove the absolute value.

$y \geq 4.5$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$y < 0$

In the piece where $y$ is negative, remove the absolute value and multiply by $- 1$ .

$- y \geq 4.5$

Write as a piecewise.

${\begin{cases} y \geq 4.5 & y \geq 0 \\ - y \geq 4.5 & y < 0 \end{cases}$

Find the intersection of $y \geq 4.5$ and $y \geq 0$ .

$y \geq 4.5$

Divide each term in $- y \geq 4.5$ by $- 1$ and simplify.

Divide each term in $- y \geq 4.5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} \leq \frac{4.5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} \leq \frac{4.5}{- 1}$

Divide $y$ by $1$ .

$y \leq \frac{4.5}{- 1}$

Simplify the right side.

Divide $4.5$ by $- 1$ .

$y \leq - 4.5$

Find the union of the solutions.

$y \leq - 4.5$ or $y \geq 4.5$

The result can be shown in multiple forms.

Inequality Form:

$y \leq - 4.5 or y \geq 4.5$

Interval Notation:

$(- \infty, - 4.5] \cup [4.5, \infty)$

Solve the Absolute Value Inequality for y |y|>=4.5

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